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# Binomial Distributions

## Equal success for independent repeated trials

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Binomial Experiments

Binomial experiments are very popular for studies because the probability of one possibility or the other can be calculated quickly and accurately. How do you identify a binomial experiment? Can an experiment that is not binomial be easily converted into a binomial experiment?

Look to the end of the lesson for the answer.

### Binomial Experiments

Binomial experiments give rise to binomial random variables, which will be the topic of our next couple of lessons. A binomial experiment is a very specific type of experiment. In order to be a binomial experiment, there are four qualifications that the experiment must meet:

1. There must be a fixed number of trials. The experiment cannot just be to roll a die until you get a 2, because the number of rolls (trials) is not fixed.
2. Each trial must be independent of the others. You cannot have a situation like “If you flip a coin and get heads, flip twice more, and if you get tails, flip three more times”.
3. Each trial must have a “success” and a “failure”. Depending on the trial, these may be identified as “yes” and “no” or “0” and “1” or “black” and “white”, etc. However, from a statistics standpoint, the outcome you are studying is generally called the “success” and the other is called “failure”.
4. The probability of success must be the same for all trials. The experiment cannot be 10 trials of pulling and keeping a card from a deck to see how many are hearts, because the probability of getting a heart would change each trial. To make this a binomial experiment, you need to replace the card each time.

#### Identifying Binomial Random Variables

1. If a fair coin if flipped 10 times, and T\begin{align*}T\end{align*} is the number of tails, is T\begin{align*}T\end{align*} a binomial random variable?

Yes, T\begin{align*}T\end{align*} is a binomial random variable, and this is a binomial experiment. It meets all four qualifications:

1. There is a specific number of trials: 10 flips
2. Trials are independent: the outcome of one coin flip does not affect the next flip
3. There are only two possible outcomes: a “success” and “failure”. Since we are counting tails, every tails is a “success” and every heads is a “failure”
4. The probability is the same for all trials: The probability of getting tails is always 50% if flipping a fair coin

2. If Trina designates Y\begin{align*}Y\end{align*} to be the number of yellow marbles she gets during nine trials of randomly pulling 1 marble from a bag filled with marbles of various colors and returning it, is Y\begin{align*}Y\end{align*} a random variable? Is it binomial?

Yes, Y\begin{align*}Y\end{align*} is a random variable, since it is the random numerical result of a limited number of independent trials of an experiment. It is also binomial, since each of the limited trials is independent, has a success/failure (yellow/not yellow), and has the same probability of success.

#### Finding Unknown Values

If N\begin{align*}N\end{align*} is the number of nines you get when rolling two standard dice three times:

1. Is N\begin{align*}N\end{align*} a binomial random variable?
2. What are the possible values of N\begin{align*}N\end{align*} ?
3. Create a histogram or pie chart showing the probability distribution of N\begin{align*}N\end{align*}.

a. Is N\begin{align*}N\end{align*} a binomial random variable?

N\begin{align*}N\end{align*} is a binomial random variable, because it is the result of a specific and limited number of independent trials of a random process and each outcome is either nine or not nine.

b. What are the possible values of N\begin{align*}N\end{align*}?

Since you could only roll a total of 9 once each trial, N\begin{align*}N\end{align*} could be 0, 1, 2, or 3.

c. Create a histogram or pie chart showing the probability distribution of N\begin{align*}N\end{align*}

The probabilities of each of the possible values of N\begin{align*}N\end{align*} would be:

(see the lesson: Understanding Discrete Random Variables, Example C, for the calculations)

• N=0:512729\begin{align*}N=0: \frac{512}{729}\end{align*}
• N=1:192729\begin{align*}N=1: \frac{192}{729}\end{align*}
• N=2:24729\begin{align*}N=2: \frac{24}{729}\end{align*}
• N=3:1729\begin{align*}N=3: \frac{1}{729}\end{align*}

A pie chart would look like this: (note that total probability = 512729+192729+24729+1729=729729\begin{align*}\frac{512}{729} + \frac{192}{729} + \frac{24}{729} + \frac{1} {729} = \frac{729}{729}\end{align*}).

#### Earlier Problem Revisited

Binomial experiments are very popular for studies because the probability of one possibility or the other can be calculated quickly and accurately. How do you identify a binomial experiment? Can an experiment that is not binomial be easily converted into a binomial experiment?

A binomial experiment must consist of a limited number of independent trials, where each trial outcome is either a success or a failure, and each trial has the same probability of success as all other trials.

A non-binomial experiment can often be viewed as binomial by carefully stating the outcome of each trial in a binomial format. For example, a non-binomial experiment might be “Count the number of heads and tails resulting from 8 flips of a fair coin”. Viewed as a binomial experiment, the same results could be collected from “How many tails do you get by flipping a fair coin 8 times”? You could then subtract the result from 8 to get the number of “not tails”, e.g. “heads”.

### Examples

#### Example 1

Mariska spins a spinner 40 times, recording the number of 4’s she gets. Is this a binomial experiment?

Yes, this is a binomial experiment because Mariska is conducting a limited number of independent random "4" or "not 4" trials, and the probability of spinnig a "4" does not change.

#### Example 2

Heidi has a bag containing 4 blue, 3 green, 5 red, and 7 yellow marbles. She defines a trial as pulling a marble, recording the color, and replacing it. She records the number of trials it takes to pull a green marble. Is this a binomial experiment?

No, Heidi is not conducting a binomial experiment because the number of trials is not specified, she just keeps pulling until she gets a green.

#### Example 3

Evan notes that 24% of online game players he polled are between 30 and 39 years old. Evan decides to create a team of players from that age range by randomly choosing names from among those he polled, keeping each one he chooses that is in his/her 30’s. If he chooses a name only 10 times, no matter the number of players he gets, is this a binomial experiment?

No, Evan is not conducting a binomial experiment because the probability that a random player will be between 30 and 39 changes each tim ehe keeps one for his team.

### Review

For questions 1-12, state that a particular experiment is or why it is not binomial:

1. A spinner has a 35% probability of landing on blue. Let B\begin{align*}B\end{align*} be the number of blues spun in 5 spins.
2. A bag contains 6 blue, 4 green, and 3 red candies. Let G\begin{align*}G\end{align*} be the number of green candies you pull out and eat in 5 trials.
3. One trial of an experiment consists of pulling a random card from a standard deck, noting it, and replacing it, you conduct 12 trials.
4. One trial consists of pulling two cards from a standard deck, noting them, and replacing them. Let T\begin{align*}T\end{align*} be the number of trials until you pull two face cards at the same time.
5. A 20-sided die is rolled ten times, and S\begin{align*}S\end{align*} is the number of sevens rolled.
6. Assume that 15% of word game players create at least 12 words out of 50 that have more than 5 letters, and you let W\begin{align*}W\end{align*} be the number of letters in words from 20 trials of 1 game each.
7. A die is rolled 20 times. What is the probability of rolling a 1 exactly 5 times?
8. You plan on choosing students (with replacement) from a population of 28, 17 of which are Juniors. You want to know how many will have to be picked before getting a Junior.
9. A new reality show is so popular that an estimated 47% of households watch it every week. You choose 20 households at random. Let X\begin{align*}X\end{align*} be the number of households watching the show.
10. H\begin{align*}H\end{align*} is the number of heads tallied over ten flips of a fair coin.
11. F\begin{align*}F\end{align*} is the number of 5’s you roll before rolling a 6, on a standard die.
12. O\begin{align*}O\end{align*} is the number of 1’s you roll in fifteen rolls of a standard die.

To view the Review answers, open this PDF file and look for section 7.6.

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### Vocabulary Language: English

binomial experiments

Binomial experiments are experiments that include only two choices, with distributions that involve a discrete number of trials of these two possible outcomes.

binomial random variable

A binomial random variable is a type of random variable that can only be used to count whether a certain event occurs or does not occur.